Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

Abstract

We study the regularity properties of the second order linear operator in ${\mathbb{R}}^{N+1}$: \begin{equation*} \mathscr{L} u := \sum_{j,k= 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum_{j,k= 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} where $A={\left(a_{jk}\right)}_{j,k=1,\dots ,m},B={\left(b_{jk}\right)}_{j,k=1,\dots ,N}$ are real valued matrices with constant coefficients, with $A$ symmetric and strictly positive. We prove that, if the operator $\mathcal{L}$ satisfies Hörmander's hypoellipticity condition, and $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\mathcal{L}u=f$ are Dini continuous functions as well. We also consider the case of Dini continuous coefficients $a_{jk}$'s. A key step in our proof is a Taylor formula for classical solutions to $\mathcal{L}u=f$ that we establish under minimal regularity assumptions on $u$.

38 pages